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Monty Hall vs Deal Or No Deal

tomhudson (43916) writes | more than 7 years ago

Puzzle Games (Games) 8

Gee ... nobody took the bait :-)

A while back, I posted that the choice is 50-50 to either keep the original prize door, or swap with the unopened door, and *nobody* bothered to jump in.

The correct choice is to always switch. The real problem is that when people try to explain their solution, they make it overly complex (or they used it as an intro to push their particular brand of "enlighenment"). A simple "truth table" suffices.

Gee ... nobody took the bait :-)

A while back, I posted that the choice is 50-50 to either keep the original prize door, or swap with the unopened door, and *nobody* bothered to jump in.

The correct choice is to always switch. The real problem is that when people try to explain their solution, they make it overly complex (or they used it as an intro to push their particular brand of "enlighenment"). A simple "truth table" suffices.

You vs The Other Doors
===================
0 -- 0 -- 1
0 -- 1 -- 0
1 -- 0 -- 0

The same setup after removing one of the other doors that doesn't have the prize:

You vs The Remaining Door
===================
0 -- 1
0 -- 1
1 -- 0

So, what does this have to do with Deal or No Deal?

Here's the scenario - you're down to 2 suitcases - the one you picked in the beginning, and one left on-stage. One of them has $1,000,000.00, and one has $0.01.

Do you swap?

Your friends and family are giving you the following advice:

  1. The Monty Hall scenario applies - swap.
  2. The Monty Hall scenario doesn't apply - It's 50-50. Do whatever you want.
  3. The odds of you picking the $1,000,000.00 were 1/26. Swap.
  4. The odds of you avoiding picking the $1,000,000.00 up until now were 1/25 - so its 95% sure you're holdng the million - don't swap.
  5. The odds of you picking the $1,000,000.00 at any point were 1/26+1/25+1/24+ ... +1/2. You almost definitely have already picked the $1,000,000.00, and since it wasn't on-stage, you're almost certainly already holding it. Don't swap.
  6. The ods that you'll make the right decision are 50-50. Do whatever you want.
  7. The odds that you've made the right pick 25 times in a row are astronomical. Swap.

Do you swap, and why or why not?

Follow-up: Just prior, you've been offered a "deal" of $300,000.00 for your suitcase. This is well below the average expected payout of a half-million. Do you take the "deal?"

cancel ×

8 comments

Chances are 50-50 (1)

squiggleslash (241428) | more than 7 years ago | (#17709674)

In the Monty Hall example, the chances are altered in your favour (well, against your choice, but in your favour in that you know that swapping will help) in that whatever door the host opens will NOT be the one with the prize money. eg,

If the situation is:

1 2 3 L L W X

The host will open door 2. If the situation is:

1 2 3 L W L X

the host will always open door 3. The only other combination is:

1 2 3 W L L X

which is the one scenario in which you'd lose by switching, hence the odds are 2:1 in favour of it being the unopen, unpicked door/box/whatever. The only other circumstance, BTW, is:

In Deal or No Deal, that situation doesn't apply. The suitcases are not being opened by a host who knows what's in them and is deliberately not picking the most attractive suitcase. Instead, the choice truly is random (well, as random as you're going to get.) If there are three suitcases left (1 is the one you have, 2 and 3 are not...)

1 2 3 H M L L H M X

It will be you eliminating 2 or 3. You don't know which would contain what. So when you accidentally eliminate the mid-range prize, leaving only the high and low boxes, either of the following scenarios may be correct:

1 2 H L L H X

Odds are 50:50. Me, I'd probably make the deal, but then $300,000 is more than enough...

Re:Chances are 50-50 that I'll not use Preview (1)

squiggleslash (241428) | more than 7 years ago | (#17709700)

...ok, here's the comment cleaned up...

In the Monty Hall example, the chances are altered in your favour (well, against your choice, but in your favour in that you know that swapping will help) in that whatever door the host opens will NOT be the one with the prize money. eg,

If the situation is:

1 2 3
L L W
X

The host will open door 2. If the situation is:

1 2 3
L W L
X

the host will always open door 3. The only other combination is:

1 2 3
W L L
X

which is the one scenario in which you'd lose by switching, hence the odds are 2:1 in favour of it being the unopen, unpicked door/box/whatever.

In Deal or No Deal, that situation doesn't apply. The suitcases are not being opened by a host who knows what's in them and is deliberately not picking the most attractive suitcase. Instead, the choice truly is random (well, as random as you're going to get.) If there are three suitcases left (1 is the one you have, 2 and 3 are not...)

1 2 3
H M L
L H M
L M H
H L M
X

It will be you eliminating 2 or 3. You don't know which would contain what. So when you accidentally eliminate the mid-range prize, leaving only the high and low boxes, either of the following scenarios may be correct:

1 2 3
H L .
L H .
H . L
L . H
X

Odds are 50:50. Me, I'd probably make the deal, but then for me $300,000 is more than enough...

Deal or no deal. (1)

arb (452787) | more than 7 years ago | (#17710702)

If it came down to two cases and the million was still live, it is a 50/50 proposition. You are the one responsible for removing the other cases and your choices are random - unlike the Monty Hall Problem, you do no have any knowledge of where to big prize(s) is(are) so your choices do not skew the probabilities the same way.

If I was offered $300,000, I'd prolly take it - $300k is a heck of a lot more then $0.01! I'm not likely to be in the same situation ever again, so the expected value of $500k is not a realistic expectation - the outcomes are: $0.01 or $1,000,000 with equal likelihood OR a guaranteed $300,000.

Re:Deal or no deal. (1)

nizo (81281) | more than 7 years ago | (#17712058)

Yeah I would take the 300K too, since I would feel totally retarded if I didn't get the million, and I'm not greedy :-) Besides fewer people would come out of the wordwork if I won 300K on national TV, but if I won a million dollars......

Re:Deal or no deal. (1)

Shadow Wrought (586631) | more than 7 years ago | (#17712324)

What's that old saying, "300k in the hand beats a mil in the suitcase?"

For my life, 300k, afte taxes, would be enough to wipe out debt and help get a nice house with land. The mil makes yout empted to think that you have more money than you actually do. In the long run, I think I'd be better of with the 300K.

But that's me. That plus after too many discussions with my mathematician of a brother over the Monty Hall problem: I see it as proof that mathematics is an invalid system. Now if you excuse me, I have some Creationists I have to go compare notes with... *sigh*

300K? (1)

RM6f9 (825298) | more than 7 years ago | (#17717540)

Deal! I can easily triple it plus in a year and a half to 2 years...

BLEH, Someone write a damn simulation (1)

Com2Kid (142006) | more than 7 years ago | (#17719526)

This is so stupid, why doesn't someone spend the 10 seconds needed to write a simulation?

My feeling is:

You are shown two doors, told that behind one of them there is a million dollars.

Before you pick, the announcer also tells you "Oh, by the way, there is a fire door over there, see that fire door? No money behind it"

There is NO difference between this and picking a door first, THEN being told "Oh yah this door here is empty".

In essence, you are given two doors to choose from, and told that one of them has a million dollars.

Indeed, the fact that you initially choose a door is irrelivent, since you REPICK a door now. "Switching" really means pick a door, since there are only two to pick from.

One of the doors without the money was ALWAYS going to be opened, therefore it does not exist.

What if the host enumerates ALL the other doors in the building that do not have the money behind them, and tells you that they, indeed, do not have money behind them. Does that change the odds any?

The door that was to be removed is pre-known and fixed. If this was a random chance problem I could maybe, maybe see alternative lines of argument working, but it is not.

The door that was to be removed was preordained, and therefore never existed as a possibility in the first place

Ahhh probablility... (1)

SatanicPuppy (611928) | more than 7 years ago | (#17740530)

1. The Monty Hall scenario applies - swap.
Monty Hall doesn't really apply; Squiggle hit that one right on the head.

2. The Monty Hall scenario doesn't apply - It's 50-50. Do whatever you want.
Any choice between two unknowns is always 50-50, so this is completely correct.

3. The odds of you picking the $1,000,000.00 were 1/26. Swap.
Irrelevant in terms of probability at this point.

4. The odds of you avoiding picking the $1,000,000.00 up until now were 1/25 - so its 95% sure you're holding the million - don't swap.
The same (bad) math applies to the .01 dollar bag

5. The odds of you picking the $1,000,000.00 at any point were 1/26+1/25+1/24+ ... +1/2. You almost definitely have already picked the $1,000,000.00, and since it wasn't on-stage, you're almost certainly already holding it. Don't swap.
This is the same argument as #4

6. The odds that you'll make the right decision are 50-50. Do whatever you want.
This is the same as #2

7. The odds that you've made the right pick 25 times in a row are astronomical. Swap.
Doesn't actually matter. If you were calculating the probability over time, then yes, you'd have a very high probability of opening the bag with the million in it...And once you've opened the bag with the .01 in it, that probability becomes 1 in 1.

It's one of those things that people don't understand with regards to probability...If I flip an unbiased coin 1000 times, and it comes up heads every time, what are the odds that it will come up heads on the 1,001st flip? Answer: 50-50, same as every other flip. Now asking what the probability is of an unbiased coin being flipped a thousand times without coming up tails is different, but that takes into account the fact that it could come up as tails on every flip....Once you've gotten to 999, the probability is comparatively high.

Since you can't actually eliminate either bag through logic or math, the only thing to do is accept the 300k, unless you get enough of a thrill out of taking the risk that you don't care about the money...Or unless you owe a loanshark 1,000,000 and it's do or die.
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